Conflict-based Statistics∗
Tomáš Müller1 , Roman Barták1 and Hana Rudová2
1
1
Faculty of Mathematics and Physics, Charles University
Malostranské nám. 2/25, Prague, Czech Republic
{muller|bartak}@ktiml.mff.cuni.cz
2
Faculty of Informatics, Masaryk University
Botanická 68a, Brno 602 00, Czech Republic
hanka@fi.muni.cz
Introduction
Value ordering heuristics play an important role in solving various problems. They allow to choose
suitable values for particular variables to compute a complete and/or optimal solution. Usually,
problem-specific heuristics are applied because problem-independent heuristics are computationally expensive. Here we propose an efficient problem-independent approach to value selection
whose aim is to discover good and poor values. We have already applied this so called conflictbased statistics (CBS) in our iterative-forward search algorithm [7]. This combination helped us to
solve a large-scale timetabling problem at Purdue University. Here we describe a general scheme
for the conflict-based statistics and apply it to local search and iterative forward search methods.
Methods similar to CBS were successfully applied in earlier works [2, 4]. In our approach,
the conflict-based statistics works as an advice in the value selection criterion. It helps to avoid
repetitive, unsuitable assignments of the same value to a variable. In particular, conflicts caused
by this assignment in the past are memorized. In contrast to the weighting-conflict heuristics
proposed in [4], conflict assignments are memorized together with the assignment which caused
them. Also, we propose our statistics to be unlimited, to prevent short-term as well as long-term
cycles.
2
General Conflict-based Statistics
The main idea behind conflict-based statistics is to memorize conflicts and discourage their future
repetition. For instance, when a value is assigned to a variable, conflicts with some other assigned
variables may occur. This means that there are one or more constraints which prohibit the
applied assignment together with the existing assignments. A counter, tracking how many times
such an event occurred in the past, is stored in memory. If a variable is selected for an assignment
(or reassignment) again, the stored information about repetition of past conflicts is taken into
account.
Conflict-based statistics is a data structure that memorizes hard conflicts which have occurred
during the search together with their frequency and assignments which caused them. More precisely, it is an array
CBS [Va = va → ¬ Vb = vb ] = cab .
It means that the assignment Va = va caused cab times a hard conflict with the assignment Vb = vb
in the past. Note that it does not imply that these assignments Va = va and Vb = vb cannot be
∗ This work is partially supported by the Czech Science Foundation under the contract No. 201/04/1102 and by
Purdue University.
used together in case of non-binary constraints. The proposed conflict-based statistics does not
actually work with any constraint. It only memorizes the conflict assignments together with the
assignment which caused them. This helps us capture similar cases as well, e.g., when the applied
assignment violates a constraint different from the past ones, but some of the created conflicts are
the same. It also reduces the total space allocated by the statistics.
The conflict-based statistics can be implemented as a hash table. Such structure is empty in
the beginning. During computation, the structure contains only non-zero counters. A counter is
maintained for a tuple [A = a → ¬ B = b] in case that the value a was selected for the variable
A and this assignment A = a caused a conflict with an existing assignment B = b. An example of
this structure
A=a
⇒
3 × ¬ B = b,
4 × ¬ B = c,
2 × ¬ C = a,
120 × ¬ D = a
expresses that variable B conflicts three times with its assignment b and four times with its
assignment c, variable C conflicts two times with its assignment a and D conflicts 120 times with
its assignment a, all because of later selections of value a for variable A. This structure can be
used in the value selection heuristics to evaluate conflicts with the assigned variables. For example,
if there is a variable A selected and its value a is in conflict with an assignment B = b, we know
that a similar problem has already occurred 3 times in the past, and the conflict A = a can be
weighted with the number 3.
3
Conflict-based Statistics in Iterative Forward Search
The iterative forward search (IFS) algorithm [7] is based on local search methods [5]. As opposed
to classical local search techniques, it operates over a feasible, though not necessarily complete,
solution. In such a solution, some variables may be left unassigned. However, all hard constraints
on assigned variables must be satisfied. This means that there are no violations of hard constraints like in systematic search algorithms. The difference here is that the feasibility of all hard
constraints in each iteration step is enforced by the unassignments of the conflicting variables.
The algorithm attempts to move from one (partial) feasible solution to another via repetitive
assignment of a selected value to a selected variable. During each step, a variable and a value from
its domain is chosen for assignment. Even if the best value is selected (whatever ‘best’ means), its
assignment to the selected variable may cause some hard conflicts with already assigned variables.
Such conflicting variables are removed from the solution and they become unassigned. Finally,
the selected value is assigned to the selected variable.
Conflict-based statistics memorizes these unassignments together with the assignment which
caused them. Let us expect that a value v0 is selected for a variable V0 . To enforce feasibility of
the new solution, some previous assignments V1 = v1 , V2 = v2 , ... Vn = vn need to be unassigned.
As a consequence we increment the counters
CBS [V0 = v0 → ¬ V1 = v1 ], CBS [V0 = v0 → ¬ V2 = v2 ], ..., CBS [V0 = v0 → ¬ Vn = vn ].
The conflict-based statistics is being used in the value selection criterion. A trivial minconflict value selection criterion selects a value with the minimal number of conflicts with the
existing assignments. This heuristics can be easily adapted to a weighted min-conflict criterion.
Here the value with the smallest sum of the number of conflicts multiplied by their frequencies is
selected. Stated in another way, the weighted min-conflict approach helps to select a certain value
that might cause more conflicts than another value. The point is that these conflicts are not so
frequent, and therefore they have a lower weighted sum. Our hope is that it can considerably help
the search to get out of a local minimum.
4
Conflict-based Statistics in Local Search
Local search algorithms [5] (e.g., min-conflict [6] or tabu search [3]) perform an incomplete exploration of the search space by repairing an infeasible complete assignment. Unlike systematic
search algorithms, local search algorithms move from one complete (but infeasible) assignment to
another, typically in a non-deterministic manner, guided by heuristics.
In each iteration step, a new assignment is selected from the neighboring assignments of the
current assignment. A neighborhood of an assignment can be defined in many different ways,
for instance, a neighbor assignment can be an assignment where exactly one variable is assigned
differently. This way, a single variable is reassigned in each move.
From the conflict-based statistics’ point of view, we would like to prohibit a move (a selection
of a neighboring assignment) which repetitively causes the same inconsistency. An inconsistency
can be identified by a variable whose assignment becomes inconsistent with assignments of some
other variables, or a constraint which becomes violated by the move.
Simply, in each iteration step, one or more variables are reassigned. These reassignments
can cause that one or more unchanged variables become inconsistent with the new assignment.
This means that there is a constraint which was satisfied by the previous assignment and it is
violated in the new assignment. The reason is that it prohibits concurrent value assignment of
some unchanged variable(s) and some reassigned variable(s). The conflict-based statistics can
memorize this problem (i.e., unchanged variables become inconsistent) together with its reason
(i.e., reassigned variables). Moreover, we can use the same structure of counters CBS [Va = va →
¬ Vb = vb ] = cab as above.
More precisely, we have reassigned variables V1 , V2 , ... Vn and unchanged variables W1 , W2 ,
... Wm which become inconsistent. Let us expect that vi is a new value assigned to Vi and
wj is a value assigned to Wj . If a constraint between Vi and Wj becomes violated, the counter
CBS [Vi = vi → ¬ Wj = wj ] is incremented. Note also that such constraint might operate
over more than two variables and some of its variables might already be inconsistent in the prior
iteration because of another constraint.
For example, there might be values v1 and v2 assigned to variables V1 and V2 respectively. As
a consequence two constraints become inconsistent:
• the constraint C1 prohibits the assignment V1 = v1 with an existing assignments W1 = w1
and W2 = w2 ,
• the constraint C2 prohibits both assignments V1 = v1 , V2 = v2 with W3 = w3 and W4 = w4 ,
but W4 = w4 is already inconsistent because of some other constraint C3 .
Then, the following counters are incremented:
• CBS [V1 = v1 → ¬ W1 = w1 ] and CBS [V1 = v1 → ¬ W2 = w2 ] wrt. C1
• CBS [V1 = v1 → ¬ W3 = w3 ] and CBS [V2 = v2 → ¬ W3 = w3 ] wrt. C2
The conflict-based statistics can be used in the move selection criterion. For example, if there
is a reassignment Va = va contained in the move, and it causes an unchanged assignment Vb = vb
to become inconsistent, such move can be weighted by the counter CBS [Va = va → ¬ Vb = vb ].
5
Experiments
We compare various local search algorithms. Basic instance of hill climbing [5] is extended by
our conflict-based statistics and tabu search [3]. Also random walk min-conflict [6] is included in
our comparison. Some experimental results for iterative forward search can be found in [7]. We
present results achieved on a Random Binary CSP with uniform distribution [1]. A random CSP
is defined by a four-tuple (n, d, p1 , p2 ), where n denotes the number of variables and d denotes the
domain size of each variable, p1 and p2 are two probabilities. They are used to generate randomly
the binary constraints among the variables. p1 represents the probability that a constraint exists
between two different variables (tightness) and p2 represents the probability that a pair of values
in the domains of two variables connected by a constraint are incompatible (density).
Figure 1 presents the number of conflicting constraints wrt. the probability p2 representing
tightness of the generated problem CSP (100, 20, 15%, p2 ). The average values of the best achieved
solutions from 10 runs on different problem instances within the 60 second time limit are presented.
500
150
450
140
Number of conflicts
Number of conflicts
400
350
300
MCRW
250
200
HC+TS
150
100
HC
HC+CBS
MCRW
120
110
HC+TS
100
90
HC+CBS
HC+CBS
80
50
0
20%
HC
130
30%
40%
HC+TS(20)
50%
60%
70%
Tightness
HC+TS(50) HC+TS(100)
80%
90%
HC+TS(200)
70
48%
49%
MCRW(1%)
50%
Tightness
MCRW(3%)
MCRW(2%)
51%
MCRW(5%)
52%
HC
Figure 1: CSP (100, 20, 15%, p2 ), the number of conflicting constraints in the best achieved solution
within 60 seconds, average from 10 runs.
550
HC+TS
Average solution value
500
HC
450
400
350
300
MCRW
250
200
150
HC+CBS
100
50
0
10%
12%
14%
16%
18%
20%
22%
24%
26%
28%
30%
Tightness
HC+CBS
MCRW(1%)
HC+TS(20)
MCRW(2%)
HC+TS(50)
MCRW(3%)
HC+TS(100)
MCRW(5%)
HC+TS(200)
HC
Figure 2: minCSP (40, 30, 43%, p2 ), the sum of all assigned values of the best solution within 60
seconds wrt. problem tightness, average from 10 runs.
We have also solved other problems with different characteristics (sparse problems, dense problems,
larger problems) but the results were similar.
For all the compared local search algorithms, a neighbor assignment is defined as an assignment
where exactly one variable is assigned differently. Min-conflict random walk (marked MCRW(Prw ))
algorithm selects a variable in a violated constraint randomly. Its value which minimizes the
number of conflicting constraints is chosen. Furthermore, with the given probability Prw a random
neighbor is selected. For hill-climbing (marked HC) and tabu search (marked HC+TS(l), where l
is the length of the tabu list), the best assignment among all the neighbor assignments is always
selected. This means an assignment of a value to a variable which minimizes the number of
conflicting constraints. Moreover, in tabu-search, if such an assignment is contained in the tabu
list (a memory of the last l assignments), the second best assignment is used and vice versa
(except of an aspiration criteria, which allows to select a tabu neighbor when the best ever found
assignment is found). For the conflict-based statistics (marked HC+CBS), the best assignment is
always selected as well, but the newly created conflicts are weighted by the frequencies of their
previous occurrences.
As we can see from Figure 1, the conflict-based statistics approach is much better than the
min-conflict random walk and the hill-climbing algorithms and slightly better than the tabu-search
on the tested problem. Moreover, there is no algorithm specific parameter (which usually depends
on the solved problem) unlike in the compared methods (e.g., the length of the tabu-list).
For the results presented in Figure 2, we turned the random CSP into an optimization problem.
The goal is to minimize the total sum of values for all variables. Note that each variable has d
generated values from 0, 1, ...d − 1. For the comparison, we used CSP (40, 30, 43%, p2 ) problems
with the tightness p2 . The problems were chosen such that each algorithms was able to find
a complete solution for each of 10 different generated problems within the given 60 second time
limit1 .
All algorithms can be easily adopted to solve this minCSP problem by selecting an assignment
with the smallest value among the neighbors minimizing the number of conflicts. But, the conflictbased statistics can do better. Here, we can add the value of the assignment to the number of
conflicts (weighted by CBS). Then, a neighbor assignment with the smallest sum of the values
and the conflicts weighted by their previous occurrences is selected. We can afford this approach
because the weights of repeated conflicts are being increased during the search, and the algorithm
is much more likely to escape from a local minima than the other compared algorithms.
For this problem, the presented conflict-based statistics was able to give much better results
than other compared algorithms. The algorithm is obviously trying to stick much more with the
smallest values than the others, but it is able to find a complete solution since the conflict counters
are rising during the search. Such behavior can be very handy for many optimization problems,
especially when optimization criteria (expressed either by some optimization function or by soft
constraints) go against the hard constraints.
6
Conclusions and Future Work
We presented a general schema for value selection heuristics called conflict-based statistics. We
applied it in iterative forward search and local search algorithms. Study of iterative forward search
was already presented in our former paper [7]. Here we concentrated on the local search extension
and we presented new experimental results with promising behavior for our new approach.
Our future study will include comparison of the described approaches on the various sets of
experimental problems. We will also analyze complexity of particular extensions. As another
step towards generalization of the conflict-based statistics, we would like to explore its possible
extension for systematic search algorithms.
References
[1] Christian Bessière. Random uniform csp generators, 1996. http://www.lirmm.fr/~bessiere/
generator.html.
[2] Rina Dechter and Daniel Frost. Backjump-based backtracking for constraint satisfaction problems. Artificial Intelligence, 136(2):147–188, 2002.
[3] Philippe Galinier and Jin-Kao Hao. Tabu search for maximal constraint satisfaction problems. In Proceedings 3rd International Conference on Principles and Practice of Constraint
Programming, pages 196–208. Springer-Verlag LNCS 1330, 1997.
[4] Narendra Jussien and Olivier Lhomme. Local search with constraint propagation and conflictbased heuristics. Artificial Intelligence, 139(1):21–45, 2002.
[5] Zbigniew Michalewicz and David B. Fogel. How to Solve It: Modern Heuristics. Springer,
2000.
[6] Steven Minton, Mark D. Johnston, Andrew B. Philips, and Philip Laird. Minimizing conflicts: a heuristic repair method for constraint satisfaction and scheduling problems. Artificial
Intelligence, 58:161–205, 1992.
[7] Tomáš Muller and Hana Rudová. Minimal perturbation problem in course timetabling. In
PATAT 2004 — Proceedings of the 5th international conference on the Practice And Theory of
Automated Timetabling, pages 283–303, 2004.
1 Except
of the hill-climbing which is not able to find a complete solution from tightness at around 16%.